Talk:Cyclic homology

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Hodge Structures and Cyclic Homology

This page should discuss results from https://faculty.math.illinois.edu/K-theory/0046/root.pdf — Preceding unsigned comment added by 128.138.65.175 (talk) 23:48, 7 September 2017 (UTC)[reply]

HC_n(A) is mistakenly used here to mean both "Hochschild complex" and "Cyclic homology" -- right?

In the section "Hints about definition", it starts off by saying that the name "HC_n(A)" denotes the nth cyclic homology. However, it later on uses the same name HC_n(A) to mean the Hochschild complex instead, and then defines the cyclic complex to be $C_{n}^{\lambda }(A):=HC_{n}(A)/\langle 1-t_{n+1}\rangle$ . I think these refer to completely different things. Svennik (talk) 15:50, 17 March 2025 (UTC)[reply]

Whoever uses wikipedia as a means for education in very high mathematics is a sucker. The article is written in an incomprehensible non-encyclopedic manner. For example, what the heck is "Hints about definition"? Now, what the heck is "Recall that the Hochschild complex groups...". Well, a reader has nowhere to "recall" from. Further, "using a notion of cyclic object" Using what? And so on. Not to say the article is severely underreferenced. I am sorely tempted to put this article for deletion.
That said, once you see a problem in wikipedia, you have to look for an answer in reliable sources cited. It is quite common that successive Wikipedians wrire somethinig without reading wtat was already written first. There is even a Russian joke with the punch line "Chukcha not reader. Chukcha writer." It looks like this is precisely what had happened here. Let us see what user:LMias (who seems to have lost interest in wp) has to say. --Altenmann >talk 18:37, 17 March 2025 (UTC)[reply]
Hi. This article was dogshit when I put my edit in, and is still dogshit obviously, with my edit I only tried to actually give a definition of cyclic homology that was absolutely not there when I first encountered this article. This is still not particularly great, and I agree there's some work to do to add sources and reword things so that the article is self-contained. But I'm not in mathematics anymore, so I won't do that :) LMias (talk) 19:47, 17 March 2025 (UTC)[reply]
Oh, and the original question: Yeah, there's a notational overlap. I think C_\lambda is pretty standard for the cyclic complex and HC is pretty standard for the Hochschild complex, but this should obviously be looked up by someone who is closer to the subject than me (who made his edit 4 years ago). LMias (talk) 19:51, 17 March 2025 (UTC)[reply]
What I "love" about maths is that every greek and latin letter has a dozen of meanings. I strongly recommend to read a brilliant spoof article Mathmanship. If you don't have a jstor account, here is a google translation of an excerpt from my Russian copy:

"But there comes a point when the reader thinks he knows all the letters. It's time to use this fact to put him down a bit. Every schoolchild knows what "pi" is, and this will help you to break away from your opponent again. The poor reader will spend a long time automatically multiplying everything by 3.1416 before he realizes that "pi" is osmotic pressure. If you are careful and don't let it slip ahead of time, it will cost him an hour and a half. The same principle can, of course, be applied to any letter. So, you can write absolutely honestly on page 141 that F is free energy, and if an astute reader is accustomed to the fact that F is free energy in the Helmholtz definition, then he will spend a lot of his own free energy on deciphering your equations before he understands that you meant the Gibbs free energy all the time, which the reader thinks is G. In general, F is a wonderful letter, it can denote not only any free energy, but also fluorine, force, farad, and also a function of an arbitrary number of real and complex variables, thereby significantly increasing the degree of chaos dS (S, as is known, denotes entropy and... sulfur)."

--Altenmann >talk 21:07, 17 March 2025 (UTC)[reply]

P.S. Cannot stop myself: the phrase "One can check that..." reminds me this one. A professor says "now it evidently follows that..." A timid voice peeps: "Sorry, er..., but how?" The prof stops for 10 minutes, then starts hastily scramble something on the board, thinks for 10 more minutes, and then triumphantly exclaims: "Of course, this is absolutely evident!" --Altenmann >talk 18:37, 17 March 2025 (UTC)[reply]